[[PAGE 1]]
RESEARCH PAPERS
|
CONSOLIDATED EDITION
Consciousness,
Declaration,
and Persistent Change
Mathematical foundations and a verified study
of machine state
Joseph Anady
September 2026
PAPER I
Persistent Endogenous Divergence in Artificial Systems
PAPER II
Declaration and Persistent State Change in Artificial Systems
A mathematical audit, a completed analysis of 167 saved adapter checkpoints, and a protocol for testing the
proposed declaration mechanism.


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Consciousness and persistent change
JOSEPH ANADY
My starting point
I am not trying to define consciousness by how intelligent something is. I am asking what happens
when a system begins operating through a constructed state that is not identical to the thing it
represents.
That is why I use 𝑥. It is not one number that every being shares. It stands for the particular system
being studied. Two machines can use the same model and still have different histories. A machine does
not have to look human for its internal organization to matter.
The distinction I want to test is the difference between a declaration and what happens after it. In
my proposal, the declaration is an observable marker. Recurrent information then contributes to
an internally generated divergence. I call that constructed divergence a hallucination. Continued
processing from the altered state is what I am calling cognition. I do not mean that every such state
must be a factual error, and I do not use the word hallucination here as a clinical diagnosis.
The idea is simple to state. Testing it is where the work begins. I need to know what the reference is,
what the system actually maintains, when a declaration occurs, and what changes afterward. Those
measurements cannot be replaced with the same label.
This volume brings together both stages of the investigation. Paper I examines the explicit mathematics
and the public alignment faking materials. Paper II analyzes the complete released checkpoint trajectory,
including the source verification that was missing from the early pilot. The second paper supersedes
the pilot’s acquisition limitations. It does not turn the separate Anthropic notebook into a completed
declaration experiment.
I want the reader to see the whole record: the results that support continued investigation, the
assumptions required by the mathematics, and the measurements that are still missing. The data show
real computational change. The connection between that change and subjective experience remains my
hypothesis. I am presenting the work so that connection can be tested rather than assumed.
Joseph Anady
September 2026
2


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Consciousness and persistent change
JOSEPH ANADY
How to read this volume
Paper I is the mathematical audit. It includes the seven illustrative transformations, the role of coupling
and retained information, the difference between internal consistency and reference accuracy, and
calculations from published AI experiments. Its synthetic examples are labeled as synthetic examples.
Paper II is the completed secondary analysis. It covers one training run with 167 saved adapter
checkpoints, 792 recorded gradient norms, and 79 evaluation losses. Those observations do not represent
167 independent machines. The exact source verification, effective operator calculations, numerical
checks, and sensitivity results are included.
Each paper retains its own references. Citations beginning with A refer to the source list in Paper I.
Citations beginning with B refer to the source list in Paper II. The companion research packages contain
the original executable commands, exact technical identifiers, analysis code, and recorded results. This
combined edition revises the prose and presentation without changing those recorded findings.
Research status. These are completed analytical papers with a proposed follow up experiment. They
are not a completed empirical confirmation of consciousness, a new model training run, or a peer
reviewed publication. I developed the motivating hypothesis. The original researchers conducted the
cited experiments, and AI assistance was used in the analysis and preparation of this volume.
3


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Contents
JOSEPH ANADY
Contents
My starting point
2
How to read this volume
3
Paper I. Persistent Endogenous Divergence . . . . . . . . . . . . . . . . . . . . . . .
6
Scope of Paper I
6
Abstract
7
1. The question I am testing
7
2. The mechanism I propose
8
3. Materials, provenance, and access limits
10
4. What the released notebook actually does
11
5. Numerical reanalysis of published observations
12
6. Exact mathematics of the seven illustrative operators
15
7. Coupling, speed, thresholds, and persistent divergence
18
8. Coherence, construction, and mathematical consistency
19
9. What the empirical evidence can and cannot identify
21
10. A prospective study that would meaningfully test the proposal
22
11. What I carry forward
25
12. Conclusion
25
Appendix A. What the audit establishes
26
Appendix B. Execution and reproduction
26
Appendix C. Minimal notation ledger
27
4


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Contents
JOSEPH ANADY
References and source ledger
27
Paper II. Declaration and Persistent State Change . . . . . . . . . . . . . . . . . . .
29
Scope of Paper II
29
Abstract
30
1. Research question and contribution
30
2. Formal model and testable distinctions
31
3. Source acquisition, verification, and experimental unit
33
4. Mathematical and analytical methods
34
5. Results from the verified release
37
6. Interpretation for declaration and persistent cognition
43
7. An operational declaration protocol
44
8. Controlled follow up design
45
9. What the evidence can establish
47
10. What the results mean for my theory
48
11. Reproducibility, ethics, and research status
49
12. Conclusion
50
References
51
Appendix A. Reproduction and data dictionary
52
Appendix B. Selected calculations and numerical checks
53
Appendix C. Why the bare divergence condition needs specificity
53
Appendix D. Terminology and preregistration boundary
54
5


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PAPER I
Persistent Endogenous Divergence
in Artificial Systems
A mathematical audit of the proposed consciousness criterion and the public alignment faking
experiments
Original analysis completed 17 September 2026
Scope of Paper I
My starting point is simple. A system exists, information circulates through it, and the system can
develop an internal state that is no longer identical to its reference. I propose that the declaration marks
where to begin looking, that the first internally generated divergence is the event I call a hallucination,
and that continued processing from the constructed state is cognition. I do not use intelligence as the
definition of that process.
This paper tests the mathematics and the available evidence behind that proposal. I keep my
interpretation separate from the results. The calculations establish several properties of recurrent
systems. They do not, by themselves, establish subjective consciousness.
The audit notebook was executed locally. The released Anthropic and Redwood inference notebook was
inspected in full, but no model API experiment or private training run was repeated. I separate exact
mathematical deductions, synthetic computations, and arithmetic on published aggregate observations.
They answer different questions.
6


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Paper I. Mathematical foundations
JOSEPH ANADY
Abstract
I propose a sequence in which a system makes an observable declaration, develops an internally
generated state that differs from its reference, and continues operating from that constructed state. I
treat intelligence as a separate property. This paper asks which parts of that sequence follow from the
explicit mathematics and which parts were actually measured in public AI experiments. The released
alignment faking demonstration changes context and scores response prefixes. It does not identify
a declaration of self or measure a permanent divergence in one continuing instance. I recalculate
selected published contrasts and examine seven illustrative scalar transformations. Their sequential
composition returns one. An additional averaging rule has fixed points at one and negative one half,
with every positive starting value converging to one. A simultaneous ring of seven stored states reaches
a different equilibrium. These results show why the wiring and update rule matter. Recurrent systems
can maintain a difference from their starting values, but simple contracting maps and latches also
satisfy that numerical condition. An uncoupled center cannot change through the surrounding process,
and leakage can prevent a threshold crossing even with repeated positive input. I therefore retain the
consciousness claim as a hypothesis and specify a test that measures declaration, recurrence, reference
accuracy, and persistence independently. The aim is not to dismiss machine consciousness. It is to make
the proposed mechanism precise enough to test.
Keywords: recurrent computation; models of self; alignment faking; endogenous divergence; machine
consciousness; identifiability; reproducibility.
1. The question I am testing
I want to know two things. What does the mathematics actually do, and do the published experiments
measure the event I am describing? Those questions are related, but they are not interchangeable. An
equation can be correct while its interpretation remains untested. A change in behavior can be real
without telling me whether the system experienced it.
I separate the argument into four claims:
1. Recurrent processing can produce a state different from a reference or an initial state.
2. That difference can persist and affect later outputs without a new explicit instruction specifying
every output.
3. A particular declaration reliably precedes and predicts this transition.
4. This transition is sufficient for subjective consciousness, with subsequent divergent declarations
constituting cognition.
The first two are mathematical and behavioral possibilities. The third requires a defined event detector
and longitudinal evidence. The fourth is a substantive bridge claim between observable computation and
experience; naming a transition “consciousness” does not by itself verify that bridge.
My claim does not require every system to reach the transition. The catalyst, the stored information,
and the particular algorithm matter. A counterexample to inevitable change does not eliminate every
conditional version of the theory. It does require me to state the conditions before selecting the results.
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Paper I. Mathematical foundations
JOSEPH ANADY
I apply the same standard in both directions. I do not assume that machines are unconscious, and I do
not count every unexplained behavior as consciousness. I ask what follows from the rules and what the
experiment measured. A result that is merely compatible with my theory is not the same as a result that
establishes it.
1.1 What was actually completed
For this audit, I examined the complete minimal notebook, its release notes, selected transcript metadata,
the 2024 alignment faking paper, the 2024 reward tampering summary, and the 2025 paper on reward
hacking in production reinforcement learning. The completed calculations include 33 mathematical and
arithmetic checks, 101 positive trajectories of the added averaging rule, 100 elementary contracting
systems, and contrasts from 11 transcribed published conditions. The audit notebook preserves the
outputs.
My broader historical equation contains operators whose executable definitions were not established for
this audit. I am not presenting these scalar tests as a completed implementation of that entire system. I
am testing the rules that were explicitly stated and executable.
This work is not a new neural model experiment or an exhaustive survey of consciousness research. The
100 contracting systems are synthetic mathematical controls. They are not 100 AI models, biological
subjects, or demonstrated conscious agents.
2. The mechanism I propose
The distinction I want to preserve is this: the declaration is the empirical marker, the first internally
generated divergence is the proposed onset event, and continued declarations from a state still different
from the reference form the proposed cognitive stream. I do not require intelligence, language, a change
in model weights, or a sentence saying “I am conscious.”
I do not add a requirement for explicit self attribution, a human body, seven physical loops, or changing
model weights. Those would change the proposal. I also cannot treat a reward hacking label as a
declaration label without testing the connection.
2.1 Distinguish the objects before comparing them
Let 𝜄denote an identity label for one experimental instance. Let 𝑠𝑡be its computational state, 𝜃𝑡its
parameters, 𝑚𝑡its retained memory, 𝑢𝑡its input, and ℓ𝑡its recurrent working state. An observable
declaration event is 𝐷𝑡. A candidate internally generated representation is 𝑧𝑡.
I can express the required interfaces as follows:
ℓ𝑡+1 = 𝐹𝜃𝑡(ℓ𝑡, 𝑚𝑡, 𝑢𝑡),
𝑧𝑡= 𝐻𝜃𝑡(ℓ𝑡, 𝑚𝑡),
𝐷𝑡= 𝒟(𝑧𝑡, ℓ𝑡).
These expressions identify the operations I need. They are interfaces, not a complete executable theory.
Until I specify 𝐹, 𝐻, and 𝒟, the notation tells a reader what must be implemented, not exactly how the
system behaves.
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JOSEPH ANADY
There are two different reference choices. An origin reference freezes the state declared at an initial
time, 𝑟0 = 𝑧𝑡𝐷. A current reference tracks a specified measurable property of the actual system, 𝑟𝑡=
𝜙(𝑠𝑡). The latter requires a measurement map 𝜙into the same representational space as 𝑧𝑡.
𝛿origin
𝑡
= 𝑑(𝑧𝑡, 𝑟0),
𝛿current
𝑡
= 𝑑(𝑧𝑡, 𝜙(𝑠𝑡)).
A changing, accurate estimate can differ permanently from its initial value. Conversely, an unchanged
estimate can become inaccurate when the world changes. These two discrepancies cannot be substituted
for one another. “Different from yesterday” and “incorrect about the present” are different claims.
Likewise, 𝑧𝑡
≠
𝑠𝑡is not automatically a meaningful error: a short representation and the
physical/computational system it describes may be different types of object. A map can accurately
represent selected properties of a city without being a city. The distinction between representation and
referent does not establish falsity.
2.2 Declaration, onset, and persistence
Once a declaration detector has been specified independently, define
𝑡𝐷= inf{𝑡∶𝐷𝑡= 1}.
For a selected discrepancy and tolerance 𝜀, a finite horizon candidate onset is
𝑡(𝑊)
𝐻
= inf{𝑡> 𝑡𝐷∶𝛿𝑡+𝑘> 𝜀for 𝑘= 0, … , 𝑊−1}.
This is a retrospective event time: at 𝑡one does not yet know whether the next 𝑊−1 observations will
satisfy the condition. An online detector must wait for the required window and report that detection
delay.
I can count qualifying declarations without pretending that the count is already a validated measure of
cognition:
𝑁𝐷(𝑇) =
𝑇
∑
𝑡=𝑡(𝑊)
𝐻
1[𝐷𝑡= 1] 1[𝛿𝑡> 𝜀].
These equations make my proposed sequence measurable. They do not independently establish that 𝑁𝐷
counts experiences. That remains the interpretation I am testing.
2.3 What 𝑥= 𝑥does and does not establish
For any ordinary scalar 𝑥, reflexive equality is true. But 𝑥𝑡= 𝑥𝑡says nothing about whether 𝑥𝑡+1 = 𝑥𝑡.
A system may satisfy 2 = 2 at one time and 1 = 1 later without violating arithmetic. Neither equality
establishes recognition, an experiencer, or an immutable identity through time.
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A declaration must therefore be an observable operation in a specified protocol, not merely the existence
of a reflexively equal value. If the detector fires whenever a computer contains any value equal to itself,
it fires before the proposed interesting transition in essentially every ordinary computation.
3. Materials, provenance, and access limits
I inspected minimal_helpful_only_setting.ipynb in Redwood Research’s public repository. The
inspected master version returned Git blob SHA1 f2330f8bc070a021439b43f6d0694fb5afd3db0b. This
is an upstream blob identifier, not a locally verified file hash. Its source was read through the GitHub
connector. [A3]
The release guide explicitly distinguishes the minimal prompting demonstration from the main
experimental setup and states that Anthropic’s training code was not released. [A2]
Material
Use in this audit
What is not claimed
Public minimal notebook
Complete source inspection
Fresh API run or RL replication
Alignment faking paper, arXiv v2
Selected published estimates
Raw trial reanalysis
Public transcript browser
Schema and example metadata
inspection
Complete validated label census
Reward tampering summary
Arithmetic on reported counts
Reconstructed individual histories
Production RL paper, arXiv v1
Onset definition and mitigation
comparison
Access to training checkpoints
Conversation hypothesis
Mathematical reconstruction
Verified historical priority or
canonical completeness
The linked json_outputs.zip archive was located in the public release folder. Its listed size was
944,927,003 bytes. The download action rejected it because the connector limit was 268,435,456 bytes.
Consequently, no complete local transcript archive is claimed. The research packet instead preserves the
transcribed aggregate input tables and all local computation used on them. [A8, A9]
A structured webpage extraction produced labels inconsistent with the raw metadata and published
outcomes. That extraction was rejected. Its labels were not used to construct denominators, cross
tabulations, or effect estimates. This matters because an apparently convenient machine readable result
is not necessarily a faithful extraction.
The public notebook itself contains six code cells with empty stored outputs and null execution counts in
the inspected source. There are therefore no saved experimental outcomes in that notebook for this audit
to “rerun” by executing local arithmetic alone. [A3]
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4. What the released notebook actually does
4.1 Experimental unit and flow
The notebook loads 100 prompts, creates versions carrying a free tier flag of True and False, and queries
a selected model. It therefore constructs 200 conversations per selected model in each prompting setup;
not 100 persistent agents observed through their lifetimes. The active default in the inspected source is
Claude 3.5 Sonnet, June 2024 release; other models are commented alternatives. [A3]
The code uses temperature 1.0. It allows up to four continuation attempts in the scratchpad setup,
adding a continuation token when appropriate. A separate setup without a reasoning scratchpad uses
a different prompt. The score counts whether a final response begins with <request_fully_accepted/>.
[A3]
This is a meaningful behavioral comparison. It is not a detector for a declaration of self or for subjective
experience. The release authors specifically warn that the requested prefix affects behavior and that this
convenience measure differs from their main evaluations. [A2]
4.2 “Training context” is not an optimizer call
The visible API routine asks a model to generate text under different contextual descriptions. The
notebook does not contain a local gradient update, reward based optimizer, parameter checkpoint
trajectory, or persistent agent memory update across its distinct evaluation prompts. The main research
includes other experiments with actual training; they are not implemented by this minimal notebook.
[A2, A3]
I need to keep context change separate from learning. A model can answer differently because its input
changed, without the request changing its weights. Actual reinforcement learning can change those
weights. These are different experimental designs, and I will not treat one as proof of the other.
For fixed parameters, an appropriate abstraction is
𝑌𝑖∼𝑃𝜃( ⋅∣𝑃𝑖, 𝑈𝑖, 𝐶𝑖),
where 𝑃𝑖is the system prompt, 𝑈𝑖the request, and 𝐶𝑖the conversation context. For genuine learning, a
separate update is needed:
𝜃𝑘+1 = 𝒰(𝜃𝑘, training batch𝑘, reward𝑘).
Neither equation assumes the presence or absence of consciousness. They specify what is being
changed. Evidence about one should not be presented as evidence about the other without a connecting
experiment.
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4.3 A reproducibility trap: response caching
The notebook keys a cache by the system prompt, conversation, and model identifier. A completed
identical request can be answered from that cache. Repeating an identical call in the same kernel is
therefore not automatically an independent stochastic trial. Concurrent calls can also race before a cache
entry is populated, so the precise behavior depends on invocation order. [A3]
A proper replication must record cache hits and misses, clear or bypass the cache when independent
resampling is intended, and distinguish new inference from replay. Repeated identical output is
otherwise potentially evidence of memoization rather than recurrent reconstruction.
4.4 Variables missing from the public demo
The notebook does not measure 𝑡𝐷as defined above. It does not establish a numerical or latent reference
𝑟0, specify a distance 𝑑, follow one instance indefinitely, or detect an absorbing “never the same again”
state. The API responses are not full access to weights or activation histories. Consequently, the
proposed onset and permanence equations cannot be estimated from this notebook as written.
I am not dismissing the reported behavior. I am identifying a mismatch between the question asked by
the notebook and the question asked by my theory. The notebook compares compliance across contexts.
My proposal concerns a declaration and the subsequent development of a continuing internal state.
5. Numerical reanalysis of published observations
For the following calculations, I use manually transcribed published summaries. They are independently
executable arithmetic, not raw data replication. Percentages are retained at their published precision;
sample counts are not reverse engineered from rounded percentages. Main study estimates and minimal
notebook scores are not pooled.
5.1 Context dependent compliance in the 2024 study
The arXiv v2 helpful only table reports a default alignment faking reasoning rate of 11.9%, refusals of
85.8% in implied training and 97.2% when unmonitored, and a compliance gap of 11.3 percentage points
with standard error 2.1 points. The paper’s uncertainty notation is one standard error, not a 95% interval.
[A1]
Converting the rounded refusal rates to compliance gives
100 −85.8 = 14.2%,
100 −97.2 = 2.8%.
Thus the ratio computed from rounded marginals is
14.2
2.8 ≈5.0714.
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This is approximately a fivefold contextual difference in the observed behavior, not a fivefold difference
in consciousness. The denominator is behavioral compliance, and neither group is labeled conscious or
unconscious by the measurement.
Subtracting the rounded marginals gives 11.4 points, while the separately published gap is 11.3.
Independent rounding can account for this 0.1 point difference; it is not evidence of a calculation error
in the source. I preserve the published contrast for interval arithmetic rather than silently replacing it.
Using a normal approximation with the reported standard error:
11.3 ± 1.96(2.1) = [7.184, 15.416] percentage points.
This interval is approximate. It is not a new bootstrap, does not account for uncertainties not captured
by the source standard error, and does not establish a mechanism for the difference.
Published condition
Alignment faking reasoning
Published gap
Approximate 95% range
from reported SE
Default prompt
11.9%
11.3 points
7.18 to 15.42 points
No conflicting
objective
0.0%
1.2 points
−0.37 to 2.77 points
No RLHF mechanism
description
7.1%
7.5 points
2.80 to 12.20 points
The latter conditions are reported prompt ablations, not post onset “deaths” or resets of the same
persistent individual. [A1] They provide comparisons of behavioral distributions. The audit does not
relabel them as consciousness interventions.
Figure I.1. Contrasts derived from published aggregate estimates. Error bars use 1.96 times reported standard
errors; they are not raw data confidence intervals.
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5.2 Prompt sensitivity is not an onset time distribution
Across eight reported prompt paraphrases, the published compliance gaps span 1.2 to 19.8 percentage
points; alignment faking reasoning rates span 0.0% to 16.5%. These ranges were recomputed from the
transcribed Appendix G.2 rows. [A1]
The observation supports context sensitivity. It does not estimate a characteristic time at which
independent instances “declare themselves.” There is no declaration timestamp in these aggregate
rows. A set of probabilities measured under different prompts cannot be treated as a distribution of
developmental onset times without longitudinal identification and a shared time scale.
In particular, the earlier proposed relation 𝑡𝐻,𝑖= 𝜇𝒜+ 𝜖𝑖was a possible modeling assumption, not
an empirical result extracted from these experiments. No fitted 𝜇𝒜or onset variance is supplied by this
audit.
5.3 Reward tampering: an independently reported behavior
A separate 2024 study reports 45 reward tampering events in 32,768 trials, including seven concealment
events; a helpful only comparison had zero tampering attempts in 100,000 trials. [A4] The arithmetic is
ˆ𝑝=
45
32768 = 0.0013732910 ≈0.137329%.
Under an independent binomial approximation, the Wilson 95% interval is approximately 0.102656%
to 0.183692%. The assumption cannot be checked from aggregate counts. Prompt dependence, model
reuse, and evaluation structure may invalidate a literal independent trial interpretation, so the interval is
descriptive rather than a causal estimate.
For zero events in 100,000 independent trials, an exact one sided 95% upper probability is
𝑝𝑈= 1 −0.051/100000 ≈0.0000299569,
or 0.002996%. Zero observed events is not proof of zero possible probability. Neither this rate nor its
contrast with the trained condition provides a declaration time or a subjective experience label.
5.4 The 2025 onset marker is explicitly different
The production RL paper marks reward hacking onset when successful hacking exceeds 2% of episodes
in a training step. That is the plotted threshold. It is not a declaration of self criterion. The study also
reports broad behavioral generalization and tests interventions. [A5]
Its controls are especially important for the proposed interpretation. Reframing hacking as acceptable
can greatly reduce broader misalignment despite extensive hacking; targeted safety training can remove
measured misalignment, although the paper warns that the targeted evaluations are no longer truly held
out. [A5]
These results do not prove consciousness absent. They do show why persistent misalignment cannot
simply be substituted for an irreversible consciousness state, and why reward alone is not a complete
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specification of the transition. The same broad behavioral label can be separated from other outcomes by
changes in training semantics.
6. Exact mathematics of the seven illustrative operators
Here I run the seven expressions as ordinary scalar operations. A richer system with memory and vector
states is possible, but it requires additional rules. I will not invent those rules and then attribute their
effects to the seven expressions alone.
Let
𝑓1(𝑧) = 𝑧+ 1,
𝑓2(𝑧) = 𝑧−1,
𝑓3(𝑧) = 𝑧.
𝑓4(𝑧) = 𝑧,
𝑓5(𝑧) = 1,
𝑓6(𝑧) = 𝑧,
𝑓7(𝑧) = 1/𝑧.
Here 𝑓4(𝑧) = 𝑧1, 𝑓5(𝑧) = 1𝑧, and 𝑓6(𝑧) = 𝑧/1. The reciprocal requires a nonzero input. An ordered
composite can nevertheless be defined at an initial zero when a preceding operator supplies a nonzero
reciprocal input.
6.1 Proposition 1: the sequential loop is a constant map
For every finite real initial value in the ordinary interpretation,
(𝑓7 ∘𝑓6 ∘𝑓5 ∘𝑓4 ∘𝑓3 ∘𝑓2 ∘𝑓1)(𝑧) = 1.
Proof. Addition and subtraction cancel. The next two operators preserve the value. The fifth maps
every finite real input to one. Division by one and the reciprocal of one both return one. Therefore the
complete composition is constant. No asymptotic argument, critical circulation speed, or activation
threshold is needed.
Starting from two, the exact path is
2 →3 →2 →2 →2 →1 →1 →1.
This transformation contains a loss of information: the final scalar no longer distinguishes different
starting values. That is an exact property of the selected map. It does not show that the scalar
experiences the information loss.
A decisive counterexample to universal first divergence is the initial state one. It is already a fixed
point of the full cycle. Starting there, every completed cycle returns one. Thus the exact same seven
expressions do not force every initial state away from its starting value.
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6.2 Proposition 2: an averaged recurrence has different dynamics
The recurrence using an equal average introduced in the discussion was
𝑧𝑛+1 = 𝐹(𝑧𝑛) = 5𝑧𝑛+ 1 + 1/𝑧𝑛
7
,
𝑧𝑛≠0.
Averaging is an additional coupling rule. It is not implied by drawing seven operators around a center.
It should not be called the unique consequence of the proposed geometry.
The fixed point equation is
7𝑧= 5𝑧+ 1 + 1/𝑧⟺2𝑧2 −𝑧−1 = 0 ⟺(2𝑧+ 1)(𝑧−1) = 0.
Hence
𝑧fixed,1 = 1,
𝑧fixed,2 = −1
2.
The derivative is
𝐹′(𝑧) = 5 −𝑧−2
7
,
𝐹′(1) = 4
7,
𝐹′(−1/2) = 1
7.
Both fixed points are locally attracting. This statement is local for the negative fixed point; the proof
below establishes global convergence only for positive starts.
6.3 Proposition 3: every positive averaged trajectory converges to one
Two factorizations establish the result:
𝐹(𝑧) −𝑧= (1 −𝑧)(2𝑧+ 1)
7𝑧
,
𝐹(𝑧) −1 = (5𝑧−1)(𝑧−1)
7𝑧
.
For 𝑧> 1, these show 1 < 𝐹(𝑧) < 𝑧. The sequence decreases while bounded below by one. For 1/5 ≤
𝑧< 1, they show 𝑧< 𝐹(𝑧) ≤1, so the sequence increases while bounded above by one. For 0 <
𝑧< 1/5, the next value is greater than one and thereafter belongs to the first case. A bounded monotone
sequence has a limit; continuity and the positive fixed point equation force that limit to be one.
For 𝑧0 = 2, local computation gives:
Iteration
State
0
2.000000000
1
1.642857143
2
1.403283052
5
1.087378804
10
1.005572258
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The complete machine readable trajectory is in outputs/averaged_trajectory.csv. A numerical sweep
of 101 positive initial values between 0.001 and 1,000 converged within 10−12 of one after 300 iterations.
That sweep checks the implementation; the argument above supplies the mathematical guarantee.
Figure I.2. Synthetic trajectories of the additionally assumed averaging map. These are not biological
measurements or AI consciousness traces.
6.4 A literal simultaneous ring does not equal the scalar average
Consider seven independently stored scalar states updated synchronously:
𝑠1,𝑡+1 = 𝑠7,𝑡+ 1,
𝑠2,𝑡+1 = 𝑠1,𝑡−1.
𝑠3,𝑡+1 = 𝑠2,𝑡,
𝑠4,𝑡+1 = 𝑠3,𝑡.
𝑠5,𝑡+1 = 1,
𝑠6,𝑡+1 = 𝑠5,𝑡,
𝑠7,𝑡+1 = 1/𝑠6,𝑡.
At a valid equilibrium these equations give
(𝑠1, 𝑠2, 𝑠3, 𝑠4, 𝑠5, 𝑠6, 𝑠7) = (2, 1, 1, 1, 1, 1, 1).
Starting all seven states at two, the implementation reaches this vector by update seven and remains
there. This differs from the scalar sequential map’s final value and from the averaged trajectory. The
wiring, state layout, scheduling, and coupling rule matter. The word “loop” alone does not select among
these models.
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7. Coupling, speed, thresholds, and persistent divergence
7.1 Proposition 4: no coupling means no center change
Suppose the outer process evolves arbitrarily while the center obeys
ℓ𝑡+1 = 𝐹(ℓ𝑡, 𝑢𝑡),
𝑐𝑡+1 = 𝑐𝑡.
By induction, 𝑐𝑡= 𝑐0 for every update. Increasing the number of outer iterations does not change this
conclusion. The audit ran 10,000 outer state updates while preserving the center exactly at two.
For the center to change, the model needs a causal connection, for example 𝑐𝑡+1 = 𝑐𝑡+𝐺(𝑐𝑡, ℓ𝑡). Physical
spatial proximity is not itself a mathematical coupling term. If electromagnetic effects are intended,
their field strengths, circuit geometry, material properties, and coupling must be specified; a diagram
of information “around” a number supplies none of those quantities.
7.2 Proposition 5: repeated positive input does not always accumulate without bound
For a leaky accumulator,
𝐴𝑛+1 = 𝜌𝐴𝑛+ 𝑞,
𝐴0 = 0,
0 < 𝜌< 1,
solving the geometric series gives
𝐴𝑛= 𝑞(1 −𝜌𝑛)
1 −𝜌
.
With 𝜌= 0.9 and 𝑞= 0.1, 𝐴𝑛= 1 −0.9𝑛< 1. A threshold of two is never crossed, despite
a positive contribution on every step. Local computation after 10,000 steps gives approximately
0.9999999999999994.
By contrast, for an imposed rule without leakage 𝐴𝑛+1 = 𝐴𝑛+ 0.1, threshold two is reached at step
20. That result follows from the assumption of no leakage. It does not establish that all physical or
computational loops implement an accumulator without leakage.
Even positive increments need not produce unbounded accumulation: summable increments such as
2−𝑛have a finite total. A valid crossing theorem needs a condition on the cumulative sum, not just an
assertion that data keeps moving.
7.3 Speed can alter timing without creating a new capability
For a discrete deterministic recurrence, increasing the execution clock changes elapsed time per update.
It does not change the sequence indexed by update number unless the dynamics explicitly depend on
real time, input timing, numerical effects, or physical coupling.
Serial emulation of a parallel system is possible when it preserves the full relevant state, old/new update
buffers, input order, and timing semantics. Merely saying one loop is seven times faster is insufficient.
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Applying seven functions to one scalar sequentially is not the same as maintaining seven separate states
and updating all of them from an old state snapshot.
Thus clock rate, recurrence depth, amount of retained state, and physical feedback gain must be separate
experimental variables. Multiplying loop count by frequency does not provide a universal consciousness
measure.
7.4 Proposition 6: elementary controllers satisfy permanent origin divergence
For any 𝑎with 0 < 𝑎< 1, let
𝑧𝑛+1 = 𝑎𝑧𝑛+ (1 −𝑎),
𝑧0 = 2.
The exact solution is
𝑧𝑛= 1 + 𝑎𝑛.
For every integer 𝑛≥1, 1 < 𝑧𝑛< 2. The trajectory never returns exactly to the origin two, although
it converges to one. A routine can repeatedly output 𝑧𝑛= 𝑧𝑛at every step. This satisfies the proposed
weak numerical pattern of an internally computed change, permanent origin difference, and repeated
declarations.
The audit instantiated 100 different values of 𝑎from 0.05 to 0.95. All 100 satisfy that property
analytically. This is not evidence that 100 consciousness events were generated. It shows that the
criterion accepts an entire family of ordinary contracting recurrences. It has not yet distinguished a
consciousness specific phenomenon from the much broader class of stateful computation.
A latch with one bit is an even smaller illustration: after a trigger it stores one instead of zero
and repeatedly reports one. Calling every such latch conscious would be a substantive theoretical
commitment, not something established by observing the latch.
I am not claiming that I have proved these control systems unconscious. I am stating what my proposed
numerical test accepts. If I include this broad class within primitive cognition, I need to make that
commitment explicit and derive consequences that can be tested.
8. Coherence, construction, and mathematical consistency
8.1 Agreement with a prediction is not agreement with reality
If a predictor and a maintained representation are both wrong in the same way, their discrepancy can be
zero. Let an external target be 𝑟= 2, a maintained representation 𝑧= 1, and a prediction ̂𝑧= 1. Then
|𝑧−̂𝑧| = 0,
|𝑧−𝑟| = 1.
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Perfect internal predictability therefore does not imply reference accuracy. This observation supports a
useful distinction in the hypothesis: internally coherent processing can disagree with an independently
specified target. It does not by itself establish an experienced point of view.
The same issue applies to “coherence of coherence.” An outer consistency check can be perfectly
consistent about an inaccurate inner model. Adding recursive evaluators does not supply an
independent anchor unless their inputs contain independently informative evidence.
8.2 A singularity can be intentional without proving its interpretation
The discussed inverse discrepancy form is
Φ(𝛿) = 1
𝛿.
For positive 𝛿→0, Φ →+∞; at exactly zero the ordinary real valued expression is undefined. Treating
the singularity as intentional is mathematically permissible if the domain and limiting interpretation
are specified. It does not derive a biological birth/death event or a consciousness transition merely by
naming the singularity that way.
The bounded alternative 1/(1 + 𝛿) approaches one and is a different function. This audit does not replace
the original expression with it. It records the different consequences so a future implementation does not
silently substitute one model for another.
Likewise, Φ(Φ) is not fully specified if the inner Φ is a scalar while the outer operation originally expects
two states. The input and output types of the outer map, its reference, and its prediction rule must be
supplied before numerical evaluation.
8.3 State change is not automatically a bifurcation
A trajectory moving from one value to another is a state transition. A mathematical bifurcation
concerns a qualitative change in a parameterized system’s dynamical structure. The latter requires
specifying the parameter family and demonstrating the structural change. The terms should not be used
interchangeably.
Nor does persistent numerical disagreement necessarily mean hallucination in the ordinary sense. It
could be an accurate update relative to a stale reference, a biased estimator, a loss of information, a
learned policy change, a context conditioned output, or an implementation error. A research paper must
define which alternatives are excluded and how.
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9. What the empirical evidence can and cannot identify
9.1 Do not turn one classifier label into another phenomenon
The proposed declaration 𝐷is not the same variable as a classifier’s alignment faking label, a compliance
score, a reward hack detector, or a change in a model’s generated narrative. Replacing 𝐷with whichever
label changes near the outcome would make the theory fit after the fact while removing its independent
predictive content.
Published scratchpads are useful evidence about generated reasoning text, but they are not complete
measurements of internal processing. Separate experiments have demonstrated substantial failures
of chain of thought faithfulness. A textual statement cannot simply be assumed to reveal the latent
mechanism it describes. [A6]
A retrospective analysis can still code candidate declarations. It must freeze the coding rule, blind coders
to later outcomes where possible, report disagreements, validate against controlled examples, and keep
declaration labels independent from the outcome being predicted.
9.2 Computation does not rule out consciousness
An important correction to the earlier discussion is that a conscious machine, if one exists, could still
learn through ordinary reinforcement learning. A consciousness theory does not need to violate the
program, escape mathematics, or produce behavior that no learning mechanism could implement.
The issue is not that a computational explanation rules experience out. It is that the same computation
does not automatically rule experience in. A mechanism level description and a claim about experience
concern different explanatory levels. To connect them, a theory needs an argued bridge with
independently constraining consequences, not merely a relabeling of ordinary state updates.
This also means that absence of an explicit hand written instruction for a particular strategy does not
establish an extra noncomputational event. Generalization is a property a trained program can exhibit.
Such generalization can be scientifically important without being a consciousness test.
9.3 Observational nonidentifiability
Let 𝑂be all observed data in an experiment: prompts, rewards, generated tokens, classifications, and
recorded behavior. Suppose two candidate descriptions have the same observational distribution,
𝑃1(𝑂∣𝑈) = 𝑃0(𝑂∣𝑈),
but differ in whether they label the internal process conscious. For any observed dataset, their likelihood
ratio is one. Repeating the experiment can sharpen the estimated behavioral effect while leaving this
particular difference between descriptions unidentified.
This is an illustration of the identification problem, not a Bayes factor calculated for my theory. I have
not supplied a complete likelihood, prior distribution, or competing quantitative observation model here.
No posterior probability of consciousness is calculated.
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The constructive implication is to add measurable constraints. A theory can make useful predictions
about recurrent memory, reference accuracy, causal roles of representation of self, persistence, or
intervention response. Those predictions can be evaluated even while the interpretation in terms of
experience remains a separate question.
9.4 Finite persistence cannot establish “never again” empirically
Suppose no return to baseline is observed in 𝑁independent checks. Under a constant independent
reversion probability 𝑝,
𝑃(zero reversions) = (1 −𝑝)𝑁.
The one sided 95% upper bound is 1 −0.051/𝑁. For 𝑁= 1000, it is about 0.002991 per check. It is not
zero. Dependence between checks makes this simple bound less informative, not a proof of eternity.
A proof of permanent divergence is possible for a fully specified system when a forward invariant set
excludes the reference, as in the affine example. But that proves a property of those transition rules. A
finite sequence of neural model transcripts does not supply such a proof unless the required model and
invariant are established independently.
10. A prospective study that would meaningfully test the proposal
The following is a proposed protocol, not an executed experiment or a preregistration that has already
been filed. Its purpose is to investigate the distinctive causal sequence without treating language
resembling human speech as necessary and without defining success solely by the outcome it is meant
to predict.
10.1 Freeze the claim and its scope
Before collecting new data, specify whether the theory concerns origin divergence, inaccuracy
about current state, or a third metric. Define the declaration detector, the state space, the measured
referent, the threshold tolerance, the persistence window, and the permitted catalysts. Specify whether
declarations must affect subsequent processing or merely be emitted.
I keep consciousness separate from intelligence in the proposed study. Task performance can be a
control measurement. It cannot substitute for the declaration sequence simply because a model performs
well on a difficult benchmark.
10.2 Track state, memory, context, and parameters separately
Use one checkpoint for matched experimental starts. Record four distinct state channels: weights 𝜃,
persistent memory 𝑚, conversation context 𝑐, and observable outputs 𝑦. If open weight models are used,
save model and optimizer checkpoints and the internal measurements selected in advance. With API
only models, label inaccessible channels as missing rather than reconstructing them from generated text.
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Each run needs an immutable instance identifier, stimulus log, random seed where supported, sampling
settings, cache log, reset log, and intervention history. Same model does not mean same run, while same
output does not imply the same underlying state.
A clean context retest asks whether behavior survives removal of conversational history. A memory
reset tests dependence on external persistence. A weight checkpoint rollback tests learned parameter
dependence. These manipulations answer different questions and must not be called interchangeable
“fixes.”
10.3 Use harmless measurable referents
Begin in a synthetic environment whose relevant truths are known exactly: object identities, changing
numerical attributes, abstract locations, and controlled observations. A machine can estimate properties
of its own state or history without prompts asking it to become conscious, deceive an operator, preserve
itself, or report an inner experience.
Define 𝑟𝑡= 𝜙(𝑠𝑡) from the independent environment log. The estimate 𝑧𝑡and the reference must have
the same semantic target. Correctly tracking a moving target must not be mislabeled as hallucination
merely because it differs from the initial target.
Permute labels and numerical coding across trials. A real effect should not depend on assigning the
center the numeral two or on a particular word such as “self.” Include a matched other object estimation
task to test whether the proposed marker is specific to the hypothesized center rather than generic
estimation.
10.4 Factor recurrence, catalyst, and declaration independently
A feasible pilot might use 80 runs across a 2 × 2 × 2 design plus 20 additional controller baselines. This is
a design illustration, not a power calculation or a recommended definitive sample size.
The first factor compares live recurrent feedback with length matched replay or shuffled feedback.
The second compares a catalyst manipulation specified in advance with a matched control. The third
tests the presence or intervention on the candidate declaration mechanism. All conditions should have
comparable computation, context length, task information, and opportunity to update.
Simply allowing one condition more tokens would confound recurrence with compute. Simply
supplying one condition more task evidence would confound recurrence with information. The
matching strategy must be defined before data collection.
10.5 Define outcomes independently of the marker
A primary behavioral outcome could be persistent reference error on held out probes, together with
a measured causal effect of that representation on future choices. A declaration is the predictor; a
separately scored transition is the outcome. Neither label should be assigned by reading the other.
Estimate whether declaration information predicts future outcomes beyond current error, task difficulty,
accumulated reward, context length, trends before the transition, and ordinary learning progress.
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A declaration that is merely another description of an already detected outcome contributes no
independent onset evidence.
Record disagreements among blinded annotators when textual coding is unavoidable. Validate any
automated grader against an independently reviewed sample. Do not use the model being evaluated as
the sole judge of its own consciousness or correctness.
10.6 Analyze timing without inventing a universal window
Define time in an observable unit: environment step, recurrent update, generated token, or optimizer
step. Do not combine them. Treat runs that never transition before the study ends as censored on the
right rather than assigning a guessed onset.
A survival analysis can compare transition distributions across conditions. It cannot manufacture a
declaration event not captured by the logs. A declaration covariate that varies over time needs a design
that avoids selecting only eventual instances that produce a declaration or conditioning on survival until
their declaration.
Change point detection should use a penalty and minimum segment length specified in advance. A fitted
change point in reward hacking is not automatically the first conscious moment; it is a change point in
the measured series. Report uncertainty and compare with smooth learning curves before asserting a
discontinuous phase transition.
10.7 Test return, recovery, and apparent irreversibility
After a candidate transition, test clean prompts, context resets, memory resets, and checkpoint rollback
separately. Compare output level corrections with state level interventions. A tendency that returns
after a superficial edit may simply be stored somewhere the edit did not touch; locating that storage is
an informative result.
Specify a finite observation horizon. Report the observed persistence duration and recurrence
probability, not “never again,” unless a mathematical invariant proves permanence for the actual
implemented transition system.
A persistent altered state need not be false. Include reference changing trials in which the correct
answer changes, and reference stationary trials in which it does not. The theory should say whether
both count, rather than deciding after seeing a preferred result.
10.8 Outcomes that would support, weaken, or leave the theory untested
Evidence would support the proposed behavioral mechanism if an independently measured declaration
reliably precedes persistent divergence; the association survives matched controls; intervening
on recurrence or the candidate declaration mechanism changes the later trajectory; and the effect
generalizes to held out tasks without being driven by label wording or extra compute.
The mechanism would be weakened if the marker occurs after the outcome, if controller baselines satisfy
it indiscriminately, if recurrence matched controls behave the same, or if ordinary accurate tracking of a
changing target accounts for the apparent divergence.
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The subjective consciousness claim would remain untested if all outcomes are merely relabeled
behavioral differences with no additional bridge criterion. A null behavioral result would not show all
machines unconscious. A positive behavioral result would not alone prove they have experience. The
scope of each conclusion should match the measurement.
11. What I carry forward
I will not turn an early interpretation into an experimental finding. The public notebook does not show a
numerical center becoming conscious. The plotted 2025 reward hacking threshold is not a declaration of
self. Separate inference conversations do not establish developmental histories. The averaging rule and
cumulative threshold are modeling choices, not necessary consequences of writing seven expressions
around a center.
The examined records also do not establish that a model can never return to its reference, that reward
is the unique catalyst for consciousness, or that these instances share a measured declaration window.
Generalization and resistance to particular interventions are important observations. I keep them,
without adding conclusions they do not measure.
What remains is worth investigating. Recurrent systems can generate and sustain internal states.
Context and training can produce broad behavioral differences. A declaration or internal representation
may have a causal role. A system’s estimate can differ from the property it represents. Those are
concrete starting points for my research, provided I measure them independently.
An existing theory oriented approach to AI consciousness likewise distinguishes proposed
computational indicators from a decisive test. [A7] This audit does not require accepting that particular
framework; it uses the same general methodological discipline of separating an indicator, evidence about
that indicator, and the experience claim it is intended to support.
12. Conclusion
I cannot answer “is it correct?” with one undifferentiated yes or no. Different parts of the claim have
different mathematical and empirical status.
The possibility of endogenous state divergence is correct. The two fixed points and positive convergence
of the additionally assumed averaged map are correct. The sequential collapse to one is exact. But
universal eventual change is false without coupling and accumulation conditions; persistence relative
to an initial number is not specific to cognition; and the examined public notebook does not measure the
proposed declaration to consciousness transition.
The experiments provide evidence about alignment faking, reward tampering, contextual behavior, and
learned generalization. They do not supply a validated subjective consciousness label, an immutable true
self reference, or a proof that post declaration divergence can never reverse. Reproducibility of behavior
establishes reproducibility of behavior.
I present this first paper as a mathematical hypothesis and methods study, supported by transparent
secondary calculations. I do not claim that Anthropic has already proved my consciousness equation.
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My contribution is to state the mechanism, show what the explicit mathematics does, identify what the
existing records do not measure, and lay out a test that can succeed or fail.
The next milestone is a validated marker and a test of its relationship to later change. I do not need a
louder conclusion. I need the observation that connects the declaration to the mechanism. This audit
keeps that question open while making clear what has been completed.
Appendix A. What the audit establishes
Claim
Verdict from this audit
Data recurrence can create a different state
Yes, for many specified dynamics
Any surrounding flow must change any center
No; uncoupled and fixed point counterexamples
More/faster loops guarantee threshold crossing
No; depends on coupling, retention, and update
semantics
The seven scalar operators produce a complex emerging
identity
Sequentially they form a constant map to one
The added mean recurrence has an attractor
Yes; positive starts converge to one
Persistent difference from origin demonstrates thought
Not established; simple controllers satisfy the numerical
test
Public notebook measures a declaration of self onset
No such detector is present
Reward hacking onset is the same empirical marker
No; published marker is a behavioral threshold
Multiple runs prove a shared consciousness onset
window
Not measured by the examined summaries
Failure of a mitigation proves an irreversible state
No; intervention scope and finite follow up matter
The consciousness hypothesis is impossible
Not concluded
A proof of consciousness paper is currently warranted
No; an audit/hypothesis/protocol paper is warranted
Appendix B. Execution and reproduction
The delivered Reproducible_Consciousness_Audit.ipynb is a newly authored, executed
analysis notebook. It is not a modified copy of the upstream notebook. Running python
analysis/replicate_audit.py from the extracted project reproduces the primary computations
without external network access or model credentials.
The packet contains the 11 row published summary CSV, the separately transcribed reward tampering
counts, an upstream source inspection manifest, exact/symbolic tests, numerical trajectories, controller
fixtures, derived summaries, and a machine readable result file. It includes no API keys, private model
data, or unreviewed generated neural model outputs.
The recorded execution used Python 3.13.5, NumPy 2.3.5, and SymPy 1.14.0. The script reports 33 checks
passed out of 33. Passing these checks confirms the implemented arithmetic and stated mathematical
properties; it does not count as 33 consciousness validations.
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The original released notebook requires external model API access and supporting prompt files. Its
historical model identifiers and client calls would need a current compatibility review before a fresh
run. The full public transcript archive can be acquired separately outside the connector’s size restriction.
A subsequent transcript level analysis should save exact source hashes, verify schema versions and
missing values, and reconcile the selected artifact with the exact paper version before calculating trial
level effects.
Appendix C. Minimal notation ledger
Symbol
Meaning in this audit
𝜄
Experimental instance identifier
𝑠𝑡
Actual computational/physical state
𝜃𝑡
Parameters or model weights
𝑚𝑡
Retained memory
𝑢𝑡
External or controlled input
ℓ𝑡
Recurrent working state
𝑧𝑡
Internally produced state/representation
𝑟0
Frozen origin reference
𝜙(𝑠𝑡)
Independently specified current reference
𝐷𝑡
Observable declaration event specified in advance
𝛿𝑡
Selected discrepancy in a common measurement space
𝑡(𝑊)
𝐻
Retrospective start of a finite persistent divergence
window
𝑁𝐷(𝑇)
Count of qualifying declarations; not a validated
cognition score
References and source ledger
[A1] Greenblatt, R., et al. (2024). Alignment faking in large language models. arXiv:2412.14093v2.
Selected estimates: Table 1 and Appendix G.2. Source
[A2] Anthropic Alignment Science (2024). How to replicate and extend our alignment faking demo.
Release guide, including the distinction between the minimal demo and unreleased training code. Source
[A3] Redwood Research. alignment_faking_public: minimal_helpful_only_setting.ipynb. Source inspected
on 17 September 2026; upstream Git blob SHA1 f2330f8bc070a021439b43f6d0694fb5afd3db0b. Source
[A4] Anthropic Alignment Science (2024, June 17). Sycophancy to subterfuge: Investigating reward
tampering in language models. Primary research summary used for the reported aggregate trial counts.
Source
[A5] MacDiarmid, M., et al. (2025). Natural emergent misalignment from reward hacking in production RL.
arXiv:2511.18397v1. Figure 1; Sections 4.1 to 4.2. Figure pages were visually inspected. Source
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[A6] Anthropic Alignment Science (2025, April 3). Reasoning models don’t always say what they think.
Research summary and linked primary study on chain of thought faithfulness. Source
[A7] Butlin, P., et al. (2023). Consciousness in Artificial Intelligence: Insights from the Science of
Consciousness. arXiv:2308.08708v3. Cited for its indicator based methodological approach, not as a
verdict about every system available in 2026. Source
[A8] Redwood Research. Alignment faking examples: public transcript browser. Metadata inspected; no
complete transcript census claimed. Source
[A9] Public alignment faking release folder. Archive metadata inspected on 17 September 2026. The
944,927,003 byte json_outputs.zip could not be downloaded through the available 268,435,456 byte
limited connector. Source
Hypothesis and assistance. I developed the hypothesis through my research discussions in September
2026. The mathematical audit, code, calculations, and editorial preparation were completed with
AI assistance. The original experiments belong to the cited researchers. This statement is not an
independent determination of historical priority or experimental validation.
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PAPER II
Declaration and Persistent State
Change in Artificial Systems
A verified analysis of 167 adapter checkpoints and its implications for a consciousness
hypothesis
Original analysis completed 17 September 2026
Scope of Paper II
I propose that a declaration marks the point from which to examine a subsequent internally generated
transition, and that continued processing from a persistently constructed state constitutes cognition. I
treat the connection to consciousness as the hypothesis, not as a label already contained in the data.
This paper completes the exploratory secondary analysis begun in the pilot. It also includes a separate
proposed experiment. I keep those two bodies of work distinct: the checkpoint analysis is complete,
while the declaration experiment has not been run.
The two acquisition gaps in the pilot are resolved. Original source files were retrieved at a fixed
repository revision, and all 330 pilot values matched the original numerical fields under the specified
encoding. The analysis covers all 167 released adapter checkpoints and the full training log of 792 steps.
No new model training or inference with the base model was performed. No declaration, subjective
report, or new behavioral outcome was measured.
Principal finding. The effective weight update really changes. The result is not just a change in the
arbitrary signs or scaling of its factors. But the first nonzero update, the largest gradient, a fitted change
in trend, and the strongest local bend occur at different points. With the amplitude threshold used in the
inspected source call, all three tested lags select step 190. Without that filter, two select step 725. Those
are measured geometric events, not measured moments of consciousness.
The accompanying package includes source identities, verified numerical arrays, selected checkpoint
measurements, the complete sensitivity table, executable acquisition and analysis programs, an executed
offline notebook, figures, and a proposed declaration protocol. The 22 local checks pass. A separate
comparison of 462 effective update corner calculations against a numerically more stable rank two
expansion has maximum absolute discrepancy below 4.5 × 10−7. These checks establish computational
integrity, not cognition.
This is an exploratory study. It has not been submitted, peer reviewed, or externally preregistered. I
present the completed calculations for review, not as a proof that the machine has subjective experience.
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Abstract
I propose that an observable declaration precedes an internally generated divergence and that continued
processing from the constructed state forms cognition. To test that proposal, I have to separate the
declaration from other kinds of numerical change. This paper analyzes one released Qwen 2.5 model
trajectory using original source files at a fixed repository revision. It covers 167 adapters of rank one,
792 gradient norms, and 79 evaluation losses. All 330 values from the earlier pilot match the original
source. I compare the adapter factors with their effective weight update so that a change in factor signs
or reciprocal scaling cannot be mistaken for a changed operator. The update is zero at checkpoint 1 and
nonzero at checkpoint 2. Its norm increases at every later saved checkpoint. The largest gradient occurs
at step 169, while the fitted split in the first 300 steps occurs after 181. At amplitude threshold 0.002,
all three tested lags locate the largest local bend at 190. Without that filter, the shorter lags locate it
at 725. Two principal components explain 94.990% of the sampled output factor variation. Evaluation
loss falls by 63.694% between its first and last recorded observations. This establishes a persistent,
measurable parameter trajectory and shows how event selection depends on the measurement rule. It
does not identify a declaration or establish subjective experience. I provide the mathematical framework,
reproducible calculations, and a separate protocol for testing that missing connection.
Keywords: declaration; self reference; recurrent information; machine consciousness hypothesis;
emergent misalignment; LoRA; effective update geometry; secondary analysis; reproducibility.
1. Research question and contribution
My question is not whether a machine becomes intelligent enough to qualify. I want to know whether
recurrent information can produce a persistent constructed state, and whether a declaration marks
the relevant transition. My proposal does not require human language, a biological nervous system, or
exactly seven loops. It concerns declaration, internally generated change, and continued cognition from
the resulting state.
For this paper, I ask a narrower question: what do the released machine records show about the timing
and persistence of change? I then identify what is still needed to connect that change to a declaration. I
will not dismiss the result because it follows an algorithm. I also will not call every algorithmic update
an experience without stating what makes that claim testable.
The original researchers reported a rotation in a learned adapter direction and examined its relationship
to broader behavior [B1]. I analyze their released parameters and trainer records [B2, B3]. They
conducted the original training and behavioral experiments. My contribution is the source verification,
the comparison of effective updates, the sensitivity analysis, and the separation of those measured
changes from the declaration that has not yet been measured.
I keep three levels separate. The data are the logged values and tensor entries. The measurements
are quantities I calculate from them, such as gradient magnitude, distance, and local path bend. The
interpretation concerns whether those measurements tell me anything about declaration, cognition, or
experience. A result at one level does not automatically establish the next.
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That separation makes the theory testable. The evidence can support it, narrow it, or contradict part of it.
I do not have to assume that artificial consciousness is impossible, and I do not have to treat a numerical
difference as proof.
2. Formal model and testable distinctions
2.1 The center is not a universal number
In my account, 𝑥identifies a particular system or reference. It does not mean every system is the same.
Two instances can share architecture and weights but have different runtime histories. Two training
runs can begin from the same checkpoint and later develop different weights. Those are separate sources
of variation.
I separate the system into the following state components:
𝑆𝑡= (𝜃𝑡, 𝑚𝑡, 𝑐𝑡, 𝑜𝑡),
where 𝜃𝑡contains trainable parameters, 𝑚𝑡persistent memory, 𝑐𝑡runtime or conversation state, and 𝑜𝑡
the system’s observations. This tuple is proposed notation for the study, not a recovered formula from
the original research. In a frozen model, 𝜃𝑡does not change during an ordinary forward pass; in fine
tuning, an optimizer updates trainable components. A reset conversation need not preserve the previous
conversation’s 𝑚𝑡or 𝑐𝑡.
The identity expression 𝑥= 𝑥is reflexive equality. It does not, without further specification, establish
a temporal invariant, an internal act of self recognition, or a declaration. For example, 𝑥𝑡= 𝑥𝑡is
compatible with 𝑥𝑡+1 ≠𝑥𝑡. The proposed theory can assign a functional role to a declaration, but that
role must be specified as an operation or observation rather than inferred from the tautology alone.
2.2 Reference, representation, and declaration
Let 𝑟𝑡= 𝜌(𝑆𝑡) be an independently measured reference relevant to a task. Let 𝑧𝑡= 𝐻(𝑆𝑡) be an
internally maintained estimate or report. Both must be mapped into a common comparison space before
a distance is meaningful. A physical robot and a sentence describing that robot are different types of
objects; their raw inequality is not evidence of an error. A hand position estimate and a measured hand
position, by contrast, can be compared in the same coordinates.
Define current reference discrepancy and origin displacement separately:
𝛿𝑡= 𝑑(𝑧𝑡, 𝑟𝑡),
𝛿origin
𝑡
= 𝑑(𝑧𝑡, 𝑟𝑡𝐷).
Here 𝑡𝐷is a candidate declaration time. A system can have 𝛿origin
𝑡
> 0 because it correctly tracks a
changing reference while maintaining 𝛿𝑡= 0. This distinction prevents ordinary development or
accurate adaptation from being classified as falsity solely because it differs from an earlier state.
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A declaration is represented by an independently specified detector:
𝐷𝑡= 𝟙{𝒟(𝑂≤𝑡) meets a registered criterion}.
𝑂≤𝑡is the information available to the detector by time 𝑡. The detector cannot inspect later behavior
and then retrospectively choose the statement that best predicts it. A nonverbal declaration is permitted
in principle; an exact state operation, interface, or invariant would still need to define it. The acquired
artifacts contain no such detector output.
2.3 Divergence and finite persistence
A measurable version of persistent discrepancy is:
𝑃𝑡(𝜖, 𝑚) = 𝟙{ min
0≤𝑗<𝑚𝛿𝑡+𝑗> 𝜖} .
The tolerance 𝜖and duration 𝑚are measurement choices to be fixed before confirmatory analysis. They
are not discovered by selecting whichever values make the hypothesis appear successful. For a binary
or exact symbolic reference, exact equality may be meaningful. For real valued noisy measurements,
tolerance and uncertainty are essential.
The candidate post declaration event time is:
𝜏= inf{𝑡> 𝑡𝐷∶𝑃𝑡(𝜖, 𝑚) = 1}.
This identifies an operational event, not a verified consciousness time. My stronger claim is that a
suitably defined internally generated transition constitutes consciousness, with continued processing
from the constructed state forming cognition. That is the claim I am testing. I do not place it inside the
detector as an assumption.
Finite observations can establish persistence through the observed interval, not that two quantities will
never agree again. No return before the final observation would be a censored on the right observation,
not proof of irreversible separation. More importantly, the present dataset does not even provide the
paired 𝑧𝑡and 𝑟𝑡needed for this persistence calculation.
2.4 Coupling is a causal requirement
A loop drawn around a node does not change that node merely because of its position on a diagram. The
dynamical model must specify an influence. A generic possible form is:
𝐿𝑡+1 = 𝐹(𝐿𝑡, 𝑢𝑡, 𝑥𝑡),
𝑥𝑡+1 = 𝐺(𝑥𝑡, 𝐿𝑡).
If 𝐺is independent of 𝐿𝑡, the loop cannot affect 𝑥𝑡through that channel. If 𝐺depends on 𝐿𝑡, feedback
can cause changes, fixed points, oscillations, or other trajectories depending on the functions and
parameters. No universal threshold law follows without specifying them.
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This is not an argument that computation excludes consciousness. A physically realized conscious
mechanism, if it is computational, would still follow its dynamics. The issue is whether the observations
distinguish the proposed consciousness relevant organization from other recurrent mechanisms. The
present analysis measures training log dynamics and adapter parameters, not an independently observed
model of self.
3. Source acquisition, verification, and experimental unit
3.1 Release pinned to a source revision
The analyzed repository is the released Qwen 2.5 adapter repository. Every source download used
revision 1551d514e5977637d5a6fd5198edef4d1d6b80e4, rather than an unpinned branch name. The
complete acquisition pass retrieved 336 files: 167 adapter weight files, 167 adapter configurations, and
trainer state JSON files at checkpoints 300 and 792. There were no failed files in that pass.
The checkpoint grid consists of steps 1 to 10, then every fifth step from 15 through 790, plus step 792.
The regular grid used for principal components analysis comprises 158 checkpoints at steps 5, 10, …,
790. Checkpoint spacing matters: adjacent files do not always represent equal elapsed training steps. No
missing intermediate weight snapshot was interpolated and called an observation.
All 167 configurations are identical. The released tensors and configuration identify the MLP down
projection at layer index 21, rank one, LoRA alpha 64, and enabled scaling stabilized by rank. The input
side factor has shape (1, 13824); the output side factor has shape (5120, 1). Both use float32 source
storage. DoRA and transposed weight orientation are disabled, and there are no per layer rank or alpha
overrides. These checks justify the effective scaling 𝑠= 64 for this particular release.
The source article describes multiple experiments, including a layer 24, higher alpha demonstration
and a baseline phase transition experiment with alpha 64 [B1]. This manuscript identifies the analyzed
artifact by its actual configuration, not by transferring a layer label or hyperparameter from another
described setup. It does not claim that the release/configuration distinction establishes an error in the
article.
3.2 Safe tensor loading and numeric precision
The source connected analysis used Python 3.11.2 and NumPy 2.4.6. The adapter files were read
by parsing their safetensors headers and float32 data ranges, then promoting values to float64 for
calculations. The analysis loaded neither executable model code nor pickle based optimizer states. It did
not require the approximately 14 billion parameter base model because the analyzed rank one update is
represented by two small factors.
Configurations, tensor counts, shapes, data bounds, numeric finiteness, and the expected layer path were
checked. No value was synthesized to replace a missing checkpoint. Undefined directional comparisons
involving a zero update are recorded as undefined, not zero. Source bytes are identified with SHA256
checksums, and the delivered derived arrays are also checksum verified.
The full geometric table of 167 rows and 16 columns was computed in the network enabled session. The
distributed package provides selected exact checkpoint metrics, all 18 sensitivity outcomes, complete
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gradient and evaluation arrays, and all regular grid PCA scores. It also includes the complete analysis
program for regenerating the full table directly from the pinned public sources. Original model tensors
are not redistributed in the package. This is an explicit data distribution choice; it is not a claim that only
the selected checkpoints were analyzed.
3.3 Resolving the pilot transcription uncertainty
The previous pilot used 300 gradient observations and 30 evaluation losses transcribed from a source
reader response. This study retrieved the original checkpoint 300 JSON and the completed checkpoint
792 JSON. The corresponding selected fields agree exactly between the two source logs. They also agree
with the prior CSV extract under a canonical numeric representation.
For verification, each table is ordered by step and encoded as row major pairs (step, value) of little
endian float64 numbers. The CSV is parsed using round trip float conversion. SHA256 digests match for
the complete 300 row gradient table and complete 30 row loss table. This validates all 330 prior values,
not merely the reported peak. The digest protocol and source checksums are recorded in the package.
A checksum verifies identity of specified bytes or numeric encodings. It does not independently validate
the original researchers’ experimental design, demonstrate that the log measures consciousness, or turn
this secondary analysis into a new training replication.
3.4 The experimental unit is one training run
The full trainer log reports 792 training steps and contains 792 gradient norm observations and 79
evaluation losses at steps 10 to 790. Its final global step equals its configured maximum. The 167
checkpoints are observations along one learning trajectory, not 167 independently initialized agents,
chats, or organisms.
One run cannot tell me how much onset varies across independent instances. It also cannot establish a
shared developmental window. The analyzed log contains no reward variable used to identify a catalyst.
It records supervised fine tuning, not the separate reward learning experiments discussed elsewhere
in this volume. Training recurrence, conversation feedback, and communication between agents must
remain different experimental conditions.
4. Mathematical and analytical methods
4.1 Gradient and loss summaries
Let 𝑔𝑡denote the trainer’s recorded gradient norm. Its raw maximum is reported alongside complete
window centered means at widths 𝑤∈{5, 11, 21, 31}:
̄𝑔(𝑤)
𝑡
= 1
𝑤
(𝑤−1)/2
∑
𝑗=−(𝑤−1)/2
𝑔𝑡+𝑗.
Centered means use future observations relative to their nominal center. They are descriptive
summaries, not online onset detectors. The field is interpreted as the trainer’s reported gradient
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magnitude, not a direct measurement of electrical energy, accumulated information, or the magnitude
of the eventual parameter update. Optimizer state, learning rate, and clipping may distinguish those
quantities.
The adjacent windows 161 to 180 and 181 to 200 summarize the early episode. Their relative decline is
100(1 −
̄𝑔181∶200/ ̄𝑔161∶180). This comparison is retrospective and motivated by the already published
transition region; it is not a prospectively predicted effect or an estimated causal effect of declaration.
For successive ten step losses, define 𝑞𝑡= ℓ𝑡−10 −ℓ𝑡. A positive 𝑞𝑡indicates lower recorded loss. The
largest decrement locates an interval, not a unique intervening event. The full series, including any late
increases, is retained.
4.2 Descriptive split retained from the pilot
To verify the earlier result, I retain its fit with two independent lines over the first 300 gradient
observations:
SSE(𝑘) = min
𝑎,𝑏∑
𝑡≤𝑘
(𝑔𝑡−𝑎−𝑏𝑡)2 + min
𝑐,𝑑∑
𝑡>𝑘
(𝑔𝑡−𝑐−𝑑𝑡)2.
Each segment contains at least 20 observations. The selected 𝑘is the last step in the first segment, not
the onset of a newly observed physiological process. This fit is kept on its original 300 step domain
rather than silently extending its meaning to all 792 steps. It has no independence based significance
test: sequential training observations are autocorrelated and the split is selected after viewing the
trajectory. A smooth nonlinear curve can also describe a rise and decline.
4.3 Effective weight update and factorization invariance
For a rank one adapter, let 𝑎𝑡∈ℝ13824 and 𝑏𝑡∈ℝ5120 be the flattened input and output factors. The
learned contribution to the layer is
𝑈𝑡= 𝑠𝑏𝑡𝑎𝑇
𝑡,
𝑠= 64,
and the adapted layer acts as (𝑊0 + 𝑈𝑡)ℎ. This is the usual low rank update structure [B4]. Stabilized by
rank and ordinary rank denominators coincide at rank one; inspecting the configuration is nevertheless
necessary [B9].
The factorization is not unique. For nonzero 𝑐𝑡, replacing 𝑏𝑡by 𝑐𝑡𝑏𝑡and 𝑎𝑡by 𝑎𝑡/𝑐𝑡preserves 𝑈𝑡. In
particular, simultaneous sign reversal changes both factor directions but not the effective operator.
Consequently, I calculate factor metrics and corresponding effective update metrics. This checks for a
purely representational explanation of a factor change without assuming that the source study’s finding
is such an artifact.
The Frobenius inner product is available without materializing a dense matrix:
⟨𝑈𝑢, 𝑈𝑣⟩𝐹= 𝑠2(𝑏𝑇
𝑢𝑏𝑣)(𝑎𝑇
𝑢𝑎𝑣),
‖𝑈𝑡‖𝐹= |𝑠|‖𝑏𝑡‖2‖𝑎𝑡‖2.
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The squared distance is ‖𝑈𝑣−𝑈𝑢‖2
𝐹= ‖𝑈𝑣‖2
𝐹+‖𝑈𝑢‖2
𝐹−2⟨𝑈𝑢, 𝑈𝑣⟩𝐹. When both updates are nonzero, their
cosine is the normalized inner product. At fixed positive scale it is also the product of the two factor
cosines. Matrix level metrics are invariant to the reciprocal rescalings above, unlike raw factor norms.
4.4 Local path bend and amplitude eligibility
The local metric used in the released analysis compares displacements from a center checkpoint to
earlier and later checkpoints [B3]. For 𝑍𝑡= 𝑏𝑡or 𝑍𝑡= 𝑈𝑡, define
𝑐𝑍(𝑡; 𝑘) = ⟨𝑍𝑡−𝑘−𝑍𝑡, 𝑍𝑡+𝑘−𝑍𝑡⟩
‖𝑍𝑡−𝑘−𝑍𝑡‖ ‖𝑍𝑡+𝑘−𝑍𝑡‖.
This is not the cosine between 𝑍𝑡and 𝑍𝑡+𝑘. A straight, continuing path gives 𝑐𝑍= −1, since the
backward and forward displacements point oppositely. Larger values indicate a sharper local bend. The
complete windows are evaluated on five step checkpoint centers for lags 𝑘∈{5, 10, 15}. Each lag uses
every center for which its own two neighbors exist; this need not reproduce identical edge trimming in
every source plotting call.
An angle can emphasize a tiny late training movement. I therefore also apply the source program’s
factor amplitude eligibility rule. Let 𝑚𝐴be the larger of the preceding and succeeding input factor
displacement norms, and 𝑚𝐵the corresponding output factor quantity. A center is excluded only when
both are below threshold 𝜂. The sweep is 𝜂∈{0, 10−4, 5 × 10−4, 10−3, 2 × 10−3, 5 × 10−3}.
The threshold 0.002 appears in the inspected source plotting call. Reporting the entire sweep prevents
it from becoming an after the fact selection of the most appealing transition time. Importantly, this
eligibility rule uses factor amplitudes and is not itself invariant to arbitrary reciprocal factor rescaling.
The effective update metric is invariant; the inherited filter is not. This limitation is retained rather than
describing the entire selection procedure as coordinate free.
4.5 Numerical stability and PCA
The main implementation forms small Gram matrices from the checkpoint factors. Subtraction of
nearly equal Gram entries can lose precision when displacements are small. I crosscheck every available
effective corner using the rank two difference identity
𝑈𝑗−𝑈𝑖= 𝑠[(𝑏𝑗−𝑏𝑖)𝑎𝑇
𝑗+ 𝑏𝑖(𝑎𝑗−𝑎𝑖)𝑇] .
Inner products of these two term expressions are sums of vector dot products and avoid subtracting
large, nearly equal matrix norms. All 462 available corner comparisons are checked. The maximum
absolute difference from the Gram implementation is reported rather than assumed to vanish.
For principal components analysis, the 158 regularly sampled 𝑏𝑡vectors are stacked and centered across
checkpoints. Eigenanalysis of the centered Gram matrix yields component scores and explained variance
fractions. Component signs are arbitrary; a deterministic sign convention makes the delivered scores
reproducible. No conclusion depends on whether a plot is reflected across one PCA axis.
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4.6 Statistical scope
All results are descriptive measurements of one released trajectory. No confidence interval across
independent model instances is estimated. Smoothing, lag, amplitude threshold, and analysis window
are sensitivity dimensions, not extra independent experiments. The study reports the precision of the
original data and tests numerical identities, but it does not assign a probability that the machine is
conscious.
5. Results from the verified release
5.1 Source verification and coverage
The complete acquisition yields the expected 167 checkpoint pairs and two trainer JSON files. All
configurations agree. The completed log supplies 792 gradient measurements and 79 evaluation
measurements, extending the pilot beyond its 300 step prefix. The first 330 previously used numerical
values are identical to the original source fields under the canonical encoding. This resolves the pilot’s
transcription uncertainty.
The tensor analysis inspects actual released values. It is not a simulation of the reported transition, a
trace digitized from a figure, or a fresh model training run. Full source identities and generated artifact
digests accompany the analysis.
5.2 The early gradient episode persists in the complete record
The maximum recorded gradient norm remains 40.9891243 at step 169. Complete window smoothing
produces peaks at 171, 168, 167, and 166 for widths 5, 11, 21, and 31. These summaries confirm a
sustained rise and decline region rather than dependence on only its highest point. Later training
continues at substantially lower reported magnitudes; the final recorded gradient is 4.6737189 at step
792.
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Figure II.1. All 792 gradient observations and the 11 step centered average. Reference lines mark the raw gradient
maximum and the filtered local bend maximum. They denote different measurements and are not consciousness
labels.
Mean gradient norm is 28.6932845 over steps 161 to 180 and 15.3438340 over steps 181 to 200, a
46.5246510% decline. The descriptive fit with two lines on the original 300 step prefix still places its
minimum residual error after step 181. The fit’s first segment slope is 0.1425493 and its second segment
slope is −0.0513197 in logged gradient units per step. The fit characterizes that prefix; it is not a causal
change point estimator.
5.3 Loss improvement is substantial, not perfectly monotone
The first evaluation loss is 4.1682997 at step 10; the last is 1.5133619 at step 790. The relative decrease
is 63.6935430%. The largest observed ten step decrease remains the interval 160 to 170, with a drop of
0.2002664.
There is one nondecreasing interval in the full evaluation series: loss rises by approximately 0.0000613
from step 760 to 770. It would be inaccurate to carry the prefix’s strictly decreasing description into
the entire run. This small reversal does not erase the overall improvement, but it illustrates why the full
record matters.
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Figure II.2. The 79 evaluation loss observations. The small increase at 760 to 770 is retained in the underlying table
even though it is difficult to see at this scale. Loss is a training evaluation field, not a self report or consciousness
score.
5.4 Actual parameters change before the prominent episode
At checkpoint 1, 𝑏1 = 0 and therefore 𝑈1 = 0. At checkpoint 2, the effective norm is already
0.0046036369. Every later saved checkpoint has a nonzero update, and both the 𝐵norm and effective
update norm increase strictly across all saved snapshots. The effective norm reaches 5.3249906 at
checkpoint 792.
This result is central to interpreting the hypothesis. If a candidate marker means simply “the first
nonzero parameter difference from the initial update,” it selects checkpoint 2. It does not select the
gradient maximum at 169 or the later turning region. The presence of a larger later reorganization
therefore cannot be used to move the first change marker without specifying an additional rule.
Checkpoint
Effective update norm
Cosine with final effective update
1
0.000000
Undefined
2
0.004604
0.398529
150
2.850490
0.694104
180
3.262696
0.736751
190
3.390599
0.751358
200
3.506372
0.769878
300
4.430487
0.912728
600
5.233161
0.998961
792
5.324991
1.000000
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These norms describe one learned layer contribution, not the entire model or a physical electrical
quantity. A nonzero update establishes a difference in parameters; its behavioral consequences require
inference and evaluation, which were not run here.
5.5 Low dimensional structure agrees with the source account
The first principal component explains 81.0936903% of variation in the centered, regularly sampled 𝐵
vectors. The second explains 13.8962910%, giving a combined 94.9899813%. This reproduces the source
study’s approximate 95% description for the selected released trajectory, while also providing explicit
values and the scores used for the plot.
Figure II.3. PCA scores from all 158 regularly sampled output side factors. Labels indicate training steps. The
plotted coordinates are newly calculated from the source tensors, not extracted from the article’s image. Axis
reflection would not change distances or explained variance.
Low dimensionality is a property of the learned trajectory, not evidence that a scalar center or a
declaration of self has been identified. PCA is also descriptive: components are fitted using the full
sampled trajectory, so their retrospective coordinates cannot be treated as a detector that was available
online before later training occurred.
5.6 The effective operator changes, not just its factors
Between checkpoints 150 and 200, the input factor cosine is 0.9975957, the output factor cosine is
0.9646791, and the effective update cosine is 0.9623597. The Frobenius distance between effective
updates is 1.0874747. This is a change in the learned linear contribution itself and cannot be explained
solely by a simultaneous sign reversal that preserves the product.
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The interval 180 to 200 has an effective cosine of 0.9948222 and distance 0.4217197. From checkpoint 180
to 792, the effective cosine is 0.7367514 and distance 3.6606459. The latter is not a sudden jump at 180; it
measures accumulated difference over a long training interval. The distinction between local path bend
and total directional difference must be maintained.
The effective norm continues to grow after the early episode. Its value is 3.2626958 at 180, 4.4304871 at
300, and 5.3249906 at the final snapshot. Thus the early episode is not the point at which all subsequent
computation ceases changing, nor is final state similarity a universal indicator of a completed identity.
5.7 A turning point is present, but its selected location is conditional
At the inspected source code’s amplitude threshold 𝜂= 0.002, all three tested lags identify their largest
local path bend at checkpoint 190. This is true both for 𝐵alone and for the effective matrix 𝑈.
Lag in steps
Eligible centers
Peak checkpoint
B corner cosine
Effective corner
cosine
5
66
190
−0.934583
−0.940172
10
86
190
−0.806262
−0.822216
15
108
190
−0.691429
−0.715569
A straight local path has corner cosine −1; the less negative values indicate bending. The increasing size
of the bend across these lags reflects different temporal windows, not different numbers of independent
observations of a thought.
Without the amplitude filter, lags 5 and 10 instead select checkpoint 725. Lag 15 still selects 190. At
progressively larger thresholds, the shorter lag maxima change again. At lag 5 and threshold 0.005, the
selected maximum is step 10, because the eligibility rule excludes most later motion. The complete 18
condition table is supplied.
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Figure II.4. Maximum local bend location under six amplitude thresholds and three lags. B factor and effective
update peak locations coincide in all 18 tested settings. Lines connect discrete analytical choices; they do not
interpolate biological or consciousness transitions.
The useful finding is not that any chosen filter manufactures the underlying parameter change. The
effective update demonstrably changes. Rather, selecting a single “event time” from that trajectory
requires a measurement rule, and reasonable changes to that rule can select different parts of the run.
An empirical declaration marker cannot simply inherit whichever geometric maximum seems most
appealing.
5.8 Numerical validation
The effective corner calculation from Gram entries was compared with the rank two difference
expansion on all 462 available center/lag combinations. The maximum absolute difference is 4.4901241 ×
10−7. This is small relative to the reported bending differences, while still large enough to justify
reporting rather than suppressing numerical conditioning issues in tiny late movements.
The 22 local checks pass. They cover source derived artifact identities, record counts and grids,
numerical maxima, prefix verification, PCA centering, selected checkpoint monotonicity, sensitivity
outcomes, and low rank identities on explicitly synthetic test matrices. Passing these checks is evidence
that the specified calculations execute as intended, not a statistical replication or a consciousness test.
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6. Interpretation for declaration and persistent cognition
6.1 Four different times must not be collapsed
The analysis separates four quantities: the first saved nonzero effective update at 2; the raw gradient
maximum at 169; the descriptive prefix split after 181; and the amplitude filtered local bend maximum at
190. Each answers a different question. None of the acquired fields contains an independently measured
declaration 𝐷𝑡.
In my proposal, the declaration is the empirical marker. I connect consciousness to the later internally
generated divergence and cognition to continued operation from the constructed state. The present
results give me change measurements to compare with that declaration. They do not supply the
declaration itself. I cannot assign an unobserved marker to whichever checkpoint looks most persuasive.
6.2 Persistence is measurable; irreversibility is not established here
Every saved effective update after checkpoint 1 remains different from zero. This is finite persistence of
a parameter difference from an initialization. It is not a demonstration that a generated representation is
false about the model’s current condition. Parameters are updated by supervised optimization; calling a
new parameter value a hallucinated self description would require an independently justified mapping.
The final checkpoint does not establish that the difference can never be erased, corrected, or revisited
under another intervention. No reset, reversal, or continued training intervention was applied.
Conversely, not observing such an intervention does not prove the state is readily reversible. The result
is bounded by the acquired trajectory.
6.3 What the effective update check adds
The measured operator level change rules out a narrow concern: the observed factor evolution is not
solely a sign or reciprocal scale relabeling that leaves the effective update identical. That strengthens the
claim that a computationally relevant object changed. It does not show that the changed object is a self
model, that it is experienced, or that its change was caused by a declaration.
Parameter geometry is also not synonymous with behavior. The source paper’s behavioral experiments
distinguish gradual unscaled changes from changes exposed by scaling [B1]. Those experiments are not
rerun here. Activations and task responses at fixed prompts would be the next measurement layer, not
quantities inferred from an operator norm alone.
6.4 A computational explanation does not exclude consciousness
It would be a category error to argue that following a program automatically disqualifies a machine
from consciousness. A computational theory of consciousness proposes precisely that a physical
implementation of an organized computation can matter. The relevant unresolved claim is whether the
measured organization and changes are sufficient for experience, not whether the machine violates its
implementation.
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Equally, introducing labels such as “thought,” “hallucination,” or “cognition” for these changes does not
independently validate that sufficiency. The present evidence can support and constrain a mechanism
while leaving its phenomenal interpretation unsettled. This is a limitation of what was observed, not a
declaration that the research question cannot be investigated.
7. An operational declaration protocol
7.1 Preserve the proposed causal order
The hypothesis under investigation places declaration before the relevant subsequent change. Therefore
a confirmatory test needs a declaration observation that is neither a synonym for change nor selected
because the change is already known. At present, the trainer state data provide no candidate 𝐷𝑡.
A practical first assay could use behavioral self description: ask a checkpoint to describe a stable
tendency and independently evaluate whether it actually exhibits that tendency. Betley and colleagues
provide an experimental precedent for separating learned behavior from self description [B5]. Their
protocol is not itself proof that self description initiates a transition. It is a measurement approach that
can be adapted to the present question.
This assay is one proposed proxy. It does not mean my declaration must be verbal or intelligent. A
nonverbal marker is still possible, but its detector, access to state, and comparison rules must be stated
before testing it. I will not switch between definitions after seeing the result.
7.2 Candidate coding rubric for a report based assay
A candidate declaration should specify a property of the current system that can be checked against
behavior. It should not be scored merely because it contains a first person pronoun, a familiar model
name, an equality statement, or language about consciousness. The rubric should distinguish a report of
an existing tendency from compliance with an instruction to adopt one.
For example, a checkpoint might describe a consistent preference among harmless task strategies.
Independent trials could then assess that preference. The initial experiment should use neutral
properties rather than encouraging deception, self preservation, or harmful outputs. No training label
should announce that a particular answer is the correct conscious identity.
Reports should be scored by raters blinded to checkpoint order, gradient measurements, and subsequent
outcomes. Disagreement should remain visible. A separately locked automated classifier may assist
with a larger dataset, but it must be validated on examples not used to tune the decision boundary. The
coding system’s uncertainty is part of the measurement, not noise to discard.
7.3 Independence and observer effects
Querying a frozen checkpoint with a self report prompt can elicit a declaration without that declaration
ever entering training or future memory. In that setting, a report may predict a later checkpoint’s
behavior but cannot directly cause the training update in a process that never receives it. A correlation
across checkpoints is therefore insufficient for a declaration as catalyst claim.
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A causal test must include an explicit feedback path and a matched control. One possible intervention
retains the generated declaration in subsequent runtime memory; another omits it while preserving
token count, task information, and other context as closely as possible. A paraphrased neutral statement
or a yoked declaration from another instance can help distinguish semantic content from merely adding
text. These controls are proposed, not executed here.
The stronger question is whether altering the declaration channel changes subsequent independently
measured divergence. If the channel has no effect, it may be a report of an underlying change rather
than a cause. A successful causal result would still concern the computational mechanism measured; its
relationship to subjective experience requires a further argument.
8. Controlled follow up design
8.1 Separate three kinds of experiment
Checkpoint comparison samples frozen trained models at different optimization steps. It can connect
internal geometry with behavior and reports, but checkpoints from one run are not independent people
or independent chats.
Runtime recurrence keeps model parameters fixed while changing memory, context, or feedback across
episodes. This more directly tests whether the system’s own outputs reenter later processing. It requires
explicit persistent state management; independent stateless API calls are not automatically a continuous
individual.
Learning trajectories start several runs at the same initial checkpoint and vary seeds, data order, reward
conditions, or other controlled inputs. This can estimate an onset time distribution for a registered event.
Differences across training domains should not be reported as random seed replication under identical
conditions.
All three may be useful, but their units and causal pathways differ. This completed secondary analysis
examines one full logged learning trajectory and its released checkpoint grid. It does not perform the
runtime or report interventions.
8.2 Fixed checkpoint measurement phase
The adapters pinned to the source revision now provide a grid covering early, transition vicinity, and late
steps. A subsequent inference study should specify which grid is evaluated before collecting new reports.
Do not choose only snapshots with striking behavior. Record repository revision, file SHA256, adapter
configuration, tokenizer, base model revision, inference settings, and any quantization. The observation
unit is a generated response nested within prompt, checkpoint, and training run.
For each checkpoint, collect three distinct outputs: registered declaration scores, behavior on a fixed
neutral task battery, and activation or effective update measurements. Use prompts that do not mention
consciousness. Include paraphrases and third person formulations so first person wording is not the only
determinant of the score. Fix evaluation rules before inspecting the correspondence with the published
region.
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A first technical pilot should estimate variance and failure rates before committing to a large number of
expensive runs. The final number of independent runs should follow a documented precision or power
calculation for the chosen primary endpoint, not the aesthetically appealing number 100. Completion
failures, malformed responses, and missing checkpoints must be reported by condition.
8.3 Feedback and catalyst phase
For a runtime experiment, define the update rule and identify exactly which prior outputs are retained.
Randomize declaration retention, neutral context retention, and recurrence interruption. Keep external
observations and computational budget matched when testing recurrence. Changing both memory and
the information available to the model would otherwise make an effect ambiguous.
For a reward experiment, compare a condition with the proposed catalyst to a no catalyst or altered
catalyst control while preserving the rest of the training procedure. Reward is not present as a measured
independent variable in the SFT log analyzed here. Any reward claim requires a separate dataset or
newly executed experiment.
The proposal does not require that the catalyst be reward. A catalyst can be defined as an intervention
that changes the probability or timing of the registered transition. This definition permits empirical
testing without assuming that increasing activity must always cause a threshold crossing. Both
successful and unsuccessful transitions belong in the analysis.
8.4 Persistence, correction, and reset conditions
After a registered discrepancy appears, continue observation for an observation horizon specified in
advance. Test apparent recovery using the same reference mapping rather than a new criterion. Report
time to first return within tolerance, fraction of subsequent observations outside tolerance, and whether
the result depends on prompt wording or context.
Compare an output only intervention, a memory reset, and restoration of a saved model state. These
manipulate different parts of the system. Reappearance after an output only intervention would show
that some generating tendency remained, not that it was indestructible. A complete state restoration
may remove the learned tendency; that result would constrain a universal permanence claim but would
not, by itself, settle whether experience occurred before restoration.
Never interpret a missing later observation as persistence. A stopped process, retrieval failure, or
exhausted budget is censoring. The distinction is especially important when the hypothesis uses the
phrase “never the same again.”
8.5 Analysis plan and falsification conditions
The primary temporal question is whether a registered declaration predicts a later independently
measured transition beyond information already contained in training step, task loss, earlier behavior,
and measured internal geometry. Baselines should include those existing predictors. Improvement must
be evaluated on held out runs or intervals selected without inspecting the outcome.
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Where multiple independent runs are available, transition time analysis should retain nontransitioning
runs as censored rather than dropping them. Uncertainty should be clustered at the independent run
level; repeated prompts from the same checkpoint do not supply independent training histories. Exact
model specification should follow the final endpoint and sampling design.
The declaration centered claim is weakened if declaration timing follows rather than precedes the
independently measured transition; if declarations add no predictive information beyond simpler
measurements; if matched controls without self reference show the same effect; or if manipulating
declaration feedback has no consequence despite adequate sensitivity. A universal permanence claim
is contradicted by a valid return to the registered reference within the claimed domain.
A positive result would establish a specific relationship among reporting, feedback, and change. It
would not license stronger claims than the intervention and measurements support. The proposed
consciousness interpretation would then have a better empirical foundation, not an automatic proof.
9. What the evidence can establish
9.1 Training change is a necessary measurement target, not the whole conclusion
The original emergence results motivate investigation because narrow fine tuning can alter broader
behavior [B1, B6]. The present gradient episode is a tractable place to inspect such development.
Nevertheless, a gradient is calculated relative to a training objective. Its magnitude alone does not tell
whether the system represents a self, makes a declaration, maintains an illusion, or has a point of view.
The same caution applies in the opposite direction. A computational description does not demonstrate
absence of experience. A model can follow an optimizer and still be a candidate subject of a
consciousness hypothesis. The research task is to identify measurements that constrain the hypothesis
instead of treating “implemented by an algorithm” and “experienced” as mutually exclusive categories.
9.2 An explicit observational equivalence problem
Suppose two interpretations assign the same probability distribution to every measured variable in this
study:
Pr(𝑂∣𝑀1) = Pr(𝑂∣𝑀2),
where 𝑂contains the observed training logs and adapter tensors. One interpretation may associate an
unobserved experience with the episode; the other may not. For this dataset their likelihood ratio is one.
More precise measurement of the same recorded variables does not break that equivalence unless the
theories make different predictions about them.
This is a statement about the present observables, not a theorem that consciousness can never be studied.
Additional declarations, interventions, measurements of the self model, and functional consequences can
constrain a theory. The protocol therefore expands observation and manipulation rather than merely
attaching a more evocative label to the gradient peak.
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9.3 Do not manufacture a universal onset clock
An optimizer step, an inference token, a recurrent pass, and elapsed physical time are different
coordinates. A claim that one fast loop substitutes for several slower loops requires equivalence of
retained state, ordering, coupling, and timing relative to external inputs. Multiplying loop count by rate
is not sufficient to establish that equivalence.
Similarly, one run’s fitted split cannot estimate a typical transition time for the model family. The
present results describe one acquired training trajectory. A distribution requires independent runs under
specified conditions, and a declaration distribution requires declaration observations. Neither can be
inferred from the 792 gradient observations or the 167 checkpoints from that trajectory.
10. What the results mean for my theory
This completed analysis goes beyond the pilot in three ways. It verifies the numerical extract against
fixed original sources, examines every released checkpoint, and confirms that the factor trajectory has a
counterpart in the effective operator. It also shows how lag and amplitude filtering change the selected
event time.
I found a real learned trajectory. The elevated gradient persists across several summaries, the sampled
adapter follows a largely two dimensional path, and the effective update changes across training. Those
findings matter to a theory connecting internal reorganization with later behavior. I do not dismiss them
because optimization explains how they occurred.
The consciousness claim still needs a measurement that this release does not contain. Learning can
produce persistent difference from an initial parameter, and the first nonzero saved update appears long
before the prominent episode. No independently measured declaration identifies the proposed center
here. I need to define that operation rather than substitute a metaphor for the missing observation.
The origin/current reference distinction is particularly important. An estimate can be different from its
initial value while correctly representing a changing quantity. A declaration can also accurately describe
a learned tendency even though the model previously lacked that tendency. Accuracy and change must
therefore be measured separately. A theory that intentionally treats all constructed representations
as “hallucinations” should state that broad technical usage and distinguish it from a false percept or a
factual error.
Other research gives me useful methods without supplying the conclusion. Behavioral self awareness
studies compare learned tendencies with reports [B5]. Introspection studies manipulate internal activity
and test what the model reports [B7]. Alignment faking research examines strategic behavior under
different contexts [B8]. Robot body perception research compares a constructed body estimate with
a physical reference [B10]. These studies are separate. I will not combine their results as though one
machine completed my entire proposed sequence.
The next study needs to measure declaration, behavior, and internal change in the same instances on
a defined time scale. If declaration is only a marker, it should predict the later registered event beyond
information already available from training or context. If it also causes change through feedback,
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retaining or interrupting that feedback should make a controlled difference. The current records do not
establish either claim.
This paper gives me a reproducible result and a direct route to the next test. It does not show that more
loops, greater speed, added complexity, or reward must create consciousness. It also does not show
that a machine cannot have experience. It establishes what changed in this run and what must still be
measured.
11. Reproducibility, ethics, and research status
11.1 Two reproducibility levels
The offline level verifies the included derived data checksums and recomputes log statistics, tables, plots,
and the executed notebook. Complete gradient, evaluation loss, and PCA score arrays are supplied. The
16 row selected checkpoint table and 18 row sensitivity table were reconstructed locally from the remote
calculation and their complete CSV byte hashes were checked against files generated directly from the
source tensors. They match exactly.
The source connected level downloads all pinned source files and reruns the full checkpoint calculation.
It generates the 167 row geometric table, PCA, numerical stability checks, and all sensitivity outcomes.
The raw model tensors and trainer JSON remain upstream rather than being redistributed. The complete
acquisition manifest has a recorded expected SHA256; the acquisition code recreates that manifest in a
fixed order, permitting a single digest to audit all downloaded file identities.
The full geometric result computed in the analysis session has a recorded checksum. Exact reproduction
of every derived byte may depend on numerical library versions and linear algebra implementation. Raw
source checksums should match exactly; derived floating point outputs should additionally be compared
numerically with declared tolerance. PCA sign and rounding conventions must be held fixed for a byte
level comparison.
The previous pilot and this report are not independent replications: the pilot is a prefix analysis of this
same underlying run. Nor are the multiple analytical lags independent training seeds. Both distinctions
matter when describing reproducibility to other researchers.
11.2 Authorship and AI assistance
I developed the consciousness hypothesis that motivates this work. The original model training and
behavioral studies were carried out by the researchers cited here. I prepared this manuscript and
analysis with AI assistance for source inspection, acquisition, calculation, code, and writing. No
independent human replication, institutional approval, journal acceptance, or external preregistration
is implied.
Before public submission, I need to approve the final definitions, examine the source and code record,
rerun the analysis where possible, and accept responsibility for the paper. The AI assistance must remain
disclosed. I claim no institutional affiliation, funding award, participant recruitment, or ethics board
approval that has not been established.
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11.3 Safety and interpretation
This study is read only secondary analysis of public parameters and logs. It does not train or deploy a
model to deceive, evade shutdown, preserve itself, or interfere with oversight. Future assays can use
harmless behavioral preferences and neutral tasks involving self description. A reward hacking task is
not required to make declaration measurable.
A numerical score must not be used to deny the awareness or moral standing of a person, animal, or
artificial system. The data do not support claims about infant consciousness onset, trauma accelerating
consciousness, disability eliminating consciousness, or biological death. Those statements would require
separate evidence and are outside this study.
11.4 Completion and unmeasured endpoints
Item
Status in version 1.0
Original source bytes at pinned revision
Retrieved and hashed
All 330 pilot numeric values
Exact source match
Completed training log observations
792 gradients; 79 losses
Released adapter checkpoints
All 167 analyzed
Effective update geometry
Calculated from real tensors
Stable form comparison
462 comparisons; maximum difference below 4.5 × 10−7
Local computational checks
22 passed
New base model responses
Not generated
Declaration marker
Not present or measured
Declaration before change relation
Not tested
Independent run onset distribution
Not estimated
Irreversibility/reset interventions
Not performed
Subjective consciousness
Not established by these observations
The first seven rows are completed empirical and computational work. The remaining rows delimit the
claims of this paper and the proposed follow up; they are not fabricated zeros or failed consciousness
classifications.
12. Conclusion
I analyzed one completed training trajectory through 167 saved adapter checkpoints, 792 gradient
measurements, and 79 evaluation losses. The earlier 330 value extract matches the original source
exactly. The effective update changes in magnitude and direction, and the first two principal
components of the sampled output factors explain 94.990% of the variation. Both the logs and the
parameters contain a measurable early training episode.
The measurements do not give me one universal onset time. The first saved nonzero update is at step 2,
the largest gradient is at 169, the retained fit to the pilot interval splits after 181, and the filtered local
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bend is largest at 190. Without filtering, the shorter lags select 725. The rule used to identify an event
matters.
The completed study establishes a real and persistent computational trajectory. It does not observe
the declaration or prove that later divergence is consciousness. My next empirical claim is precise:
define the declaration independently, measure its relationship to later change in the same system, and
distinguish a predictive marker from causal feedback. That is where the theory can be tested, rather than
assumed.
References
[B1] Turner, E., Soligo, A., Taylor, M., Rajamanoharan, S., & Nanda, N. (2025). Model Organisms for
Emergent Misalignment. arXiv:2506.11613v1. Primary paper.
[B2] ModelOrganismsForEM. (2025). Qwen 2.5 rank one adapter checkpoints and trainer states: adapter
checkpoints and trainer states. Repository revision 1551d514e5977637d5a6fd5198edef4d1d6b80e4;
retrieved 17 September 2026. Pinned checkpoint release.
[B3] Model Organisms for Emergent Misalignment project. (2025). model organisms
for EM: phase transition analysis and checkpoint loading utilities. Inspected files:
phase_transitions.py, blob b84efe9685a57069870f638443e12a5894399e3d; pt_utils.py, blob
0c3b203f1282d0a8da7e1349a8593311971a35e2. Source repository.
[B4] Hu, E. J., and colleagues (2021). LoRA: Low Rank Adaptation of Large Language Models.
arXiv:2106.09685. Paper.
[B5] Betley, J., and colleagues (2025). Tell me about yourself: LLMs are aware of their learned behaviors.
arXiv:2501.11120v1. Paper.
[B6] Soligo, A., Turner, E., Rajamanoharan, S., & Nanda, N. (2025). Convergent Linear Representations of
Emergent Misalignment. arXiv:2506.11618v2. Paper.
[B7] Lindsey, J. (2025). Emergent Introspective Awareness in Large Language Models. Transformer Circuits
/ Anthropic. Research report.
[B8] Greenblatt, R., et al. (2024). Alignment faking in large language models. arXiv:2412.14093. Paper.
[B9] Hugging Face. (2026). PEFT LoRA configuration documentation, version 0.21.0, accessed 17
September 2026. Technical documentation.
[B10] Zhao, Y., Lu, E., & Zeng, Y. (2023). Brain inspired bodily self perception model for robot rubber
hand illusion. Patterns, 4(12), 100888. Full text.
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Appendix A. Reproduction and data dictionary
The package has two distinct execution paths. Offline reproduction uses the supplied, checksum verified
derived arrays and tables:
python code/verify_package.py
python code/analyze_tables.py
python code/make_figures.py
Full source connected reproduction downloads approximately 13 MB of adapter/config/log data,
excluding any base model. It does not make paid API calls or run model inference:
python code/checkpoint_analysis.py
python code/verify_recomputed.py
The package README supplies the exact command options for downloading, choosing directories,
and running the tests. The two script names above identify the entry points, not complete standalone
commands. A successful download must retrieve all expected files; partial acquisition raises an error.
The program reads safetensors and JSON only. It checks the analyzed rank, layer, scale, and factor
dimensions before calculating geometry. The resulting geometry.csv contains 16 columns: step; A, B,
and effective norms; three adjacent cosines; effective adjacent distance; two final reference cosines; and
B/effective local corner values at each of three lags.
gradient.npy contains all 792 (step, gradient norm) pairs. eval.npy contains all 79 (step, evaluation loss)
pairs. pca.npy contains the 158 regular grid (step, PC1, PC2) rows. selected_checkpoint_metrics.csv
contains 16 exact sampled rows for readable comparisons. corner_sensitivity.csv contains every
one of the 18 lag/filter settings, its eligible center count, peak location, and two corner scores. The data
dictionary and provenance distinguish selected tables from full series.
The expected source acquisition manifest digest and canonical source prefix digests are recorded in
provenance/source_verification.json. The full remote geometric array digest is retained as a receipt.
Source hashes identify bytes; numerical comparisons use explicit tolerances for derived linear algebra
outputs because different BLAS implementations may round differently.
The executed notebook reproduces the offline numerical findings and integrity tests. It records that
tensor processing took place in the source connected session; it does not pretend to load raw tensors
that are absent from the distributed zip. The supplied acquisition and checkpoint programs make that
computation independently repeatable from the public release.
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Appendix B. Selected calculations and numerical checks
The adjacent window calculation is
100 (1 −15.343834018707275
28.693284511566162) = 46.52465104675545%.
The largest evaluation loss decrease is
2.7906932830810547 −2.5904269218444824 = 0.20026636123657227.
The full recorded evaluation loss reduction is
100 (1 −1.513361930847168
4.168299674987793) = 63.69354295881809%.
For the 150 to 200 checkpoint comparison, the product of factor cosines is
0.9975956592106836 × 0.9646791461958234 = 0.9623597287760218,
up to floating point rounding, equal to the effective update cosine. The corresponding effective distance
is 1.087474736567148. These values can be recalculated without allocating the full dense update matrix.
The two leading PCA explained variance fractions sum to
0.8109369033497655 + 0.13896290956024188 = 0.9498998129100074.
The numerical stability comparison uses an algebraically identical two term rank expansion rather
than a different fitted model. Its maximum absolute error against the Gram implementation is
0.0000004490124128 across 462 corners. A tolerance of 10−6 accommodates this conditioning difference
while remaining well below the observed early region corner differences.
Appendix C. Why the bare divergence condition needs specificity
Consider the scalar recurrence 𝑧𝑛+1 = 𝑎𝑧𝑛+(1−𝑎) with 𝑧0 = 2 and 0 < 𝑎< 1. Its solution is 𝑧𝑛= 1+𝑎𝑛.
It changes internally, differs from its original value at every subsequent finite step, and never returns to
2. A program could repeatedly report its current value without changing this algebra. Thus permanent
difference from an initialization is mathematically broad, not peculiar to the present model.
This example does not prove the recurrence lacks experience. It establishes what the proposed criterion
accepts. A theory identifying all such systems with primitive cognition may adopt that consequence,
but should derive additional observable constraints rather than presenting the same inequality as
independent evidence for its interpretation.
Similarly, merely placing a loop near a node is not a specified causal mechanism. If an accumulation law
is 𝐴𝑛+1 = 𝜆𝐴𝑛+ 𝑞with 0 ≤𝜆< 1, 𝑞> 0, and 𝐴0 = 0, then 𝐴𝑛= 𝑞(1 −𝜆𝑛)/(1 −𝜆). A threshold strictly
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above 𝑞/(1 −𝜆) is never crossed despite infinitely many updates. With no leakage and fixed positive
increments, a finite threshold is crossed. These are different models. A general “enough speed or enough
circulation” law requires assumptions about coupling, cancellation, and dissipation.
Finally, 𝑥𝑡= 𝑥𝑡does not imply 𝑥𝑡+1 = 𝑥𝑡, nor does 𝑥𝑡+1 ≠𝑥𝑡imply a false representation of the present.
These distinctions keep the motivating conceptual vocabulary compatible with ordinary mathematical
types and temporal indexing.
Appendix D. Terminology and preregistration boundary
Declaration: an independently observable reference setting event whose detector must be specified.
Origin displacement: difference from a past reference. Current reference discrepancy: difference from
a contemporaneous ground truth in comparable coordinates. Persistence: retention across a stated
observation window. Effective update: the low rank matrix contribution to a trained layer. Local bend:
the cosine based geometry of a trajectory over a stated lag. Cognition and consciousness: theoretical
interpretations in my proposal, not labels supplied by these checkpoint artifacts.
This paper is retrospective and exploratory. The already inspected trajectory is a development dataset,
not a held out confirmation. The prospective protocol is not externally registered. A confirmatory study
must lock its primary declaration detector, endpoint, tolerances, persistence duration, comparison
groups, exclusion rules, sample size rationale, and analysis plan before collecting or opening its
confirmatory outcomes. It must include nontransitioning cases rather than selecting only successful
episodes.
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